Is it possible to copy data from one scene and paste it into another within the same .blend file?
There are other ways to get data from another scene:

A) when you create a new scene, you're given several choices:



(taken from the relevant manual section)
"

New Creates an empty scene with default values.

Copy Settings Creates an empty scene, but also copies the settings from the active scene into the new one.

Link Objects This option creates a new scene with the same settings and contents as the active scene. However, instead of copying the objects, the new scene contains links to the objects in the old scene. Therefore, changes to objects in the new scene will result in the same changes to the original scene, because the objects used are literally the same. The reverse is also true.

Link Object Data Creates new, duplicate copies of all of the objects in the currently selected scene, but each one of those duplicate objects will have links to the object-data (meshes, materials and so on) of the corresponding objects in the original scene.

This means that you can change the position, orientation and size of the objects in the new scene without affecting other scenes, but any modifications to the object-data (meshes, materials, etc.) will also affect other scenes. This is because a single instance of the “object-data” is now being shared by all of the objects in all of the scenes, that link to it. This has the effect of making a new independent copy of the object-data.

Full Copy Using this option, nothing is shared. This option creates a fully independent scene with copies of the active scene’s contents. Every object in the original scene is duplicated, and a duplicate, private copy of its object-data is made as well. 

"

B) You can link any object from one scene to another.

As this other "Make Links" manual section says

"
Links objects between scenes or data-blocks of the active object to all selected objects. In some case (i.e. Object Data, Modifier) the target objects must be of the same type than the active one or capable of receiving the data. The existing data-block of which will be unlinked from them.

Objects to Scene... Lets you create links to the selected objects into a different scene than the current one. A scene name must be chosen other than that of the current one. The Link Objects to Scene Operator panel lets you choose between scenes.

This makes the same object exist in two different scenes at once, including its position and animation data. The object’s origin will change its color.

"